1) Box fans (and other axial fans like your ceiling fan) are terrible at pulling air and an HVAC filter will significantly reduce fan speed due to the added pressure differential across the filter. As a reference, a new MERV 13 filter can reduce fan speed by ~33% when mounted to the intake side, and ~66% if mounted to the front of a box fan. The motor used in your box fan is cooled by the air passing around it, which is why choking off the air flow through your fan can lead to overheating, damaging your fan and creating a fire hazard.
If you're going to make a DIY air purifier, mount the filter to the intake side. You'll get much better performance, your fan will stay cleaner, and there's less risk of damage.
2) Some analysis of the test data from the posted article: CADR stands for Clean Air Delivery Rate, measured in cfm. It's basically a measure of how efficiently ambient air in a confined space is purified. A higher CADR means particles are pulled out of the air quicker. For now, just treat this as a relative quality value. Using the PM 2.5 graph for the tiny room experiment in the article, I'd estimate that the DIY filter the author created has a CADR value of 45-50. This is a pretty low value. The room air purifier (RAP) he used is even worse. Then, from the large room graph, I'd estimate the DIY purifier as a CADR of 30-35. This is a rough estimate for two reasons. One, room size is an important variable in this calculation and I don't have an exact value. Two, when I test a purifier, the test chamber starts at a PM 2.5 around 10^5. At the extremely low starting concentrations used in the author's experiments, the percentage of particles that are removed due to natural decay is much more significant. Side note, cigarette smoke is the standard for room air purifier testing. Incense sticks are used less commonly due to their slower particle generation.
3) For wildfire season, I recommend a MERV 13 filter for overall performance, cost effectiveness, and for smoke particle capture (PM 2.5). There is a clear trend of diminishing returns for a DIY box fan filter beyond MERV 12-13 filters, peaking at a CADR value of ~150 for a MERV 13 filter for box fans I've tested at their highest fan speeds. A MERV 13 filter is about 50% better than a MERV 10 in a box fan configuration, while a MERV 14 is actually slightly worse. This will vary based on filter brands, too. Rule of thumb, quality matters. We actually rate our filters a bit lower than their actual performance for a number of reasons.
So, my company offers a standing room air purifier with a HEPA filter for about $200. It has a CADR value of 158. A box fan and a MERV 13 filter cost about $40 and has a CADR around 150 - pretty good for a DIY substitution. We even offer cheaper room air purifiers with even lower CADR values. So why buy an expensive room air purifier?
First, room air purifiers use a radial fan rather than an axial fan. Axial fans create a low pressure area on the exhaust side, drawing air through the fan. Radial fans draw in lower pressure air near the axle of the fan and push out higher pressure air at an exhaust port at the radius of the fan. Room air purifiers use a radial fan to push air through the high pressure drop HEPA filters they are designed to use. HEPA filters are qualified to remove +99.9% of tiny particles (PM 0.3) in one pass. The thick filtration media requires a high pressure differential to pull air through it. A HEPA filter on an axial box fan is going to kill the motor. If you care about PM 0.3 particles, only a HEPA filter will do the job.
Also, longevity. A room air purifier and HEPA filter should run for a year or longer without needing to change filters. An HVAC filter is meant to last 3 months in your home air system under a normal particle load. This lifespan can be much shorter due to poor conditions, such as smoke particles during wildfire season or drywall dust from a renovation project. (Side note - seriously, replace your filter after doing any drywall work. Anything better than a fiberglass filter can clog in just a day or two of dusty drywalling. Also also, fiberglass filters do absolutely nothing, don't buy them.) The room air purifier is designed to run for years with a high pressure differential filter. Your box fan is not. A cheap 20" box fan with an HVAC filter is a good temporary solution, but will not last nearly as long as a room air purifier used daily.
Last, noise and style. RAP units are very quiet and blend into your living room space. A large box fan with a strapped on furnace filter makes for an interesting conversation piece, if you could hear your guests.
4) Don't worry about hermetically sealing your 20x20 filter to the box fan. From my experience, a filter with tape sealing every side to the fan is no more efficient than a filter held to the fan at a couple of contact points. Once the fan is on, the intake air will hold the filter close to the fan. Even with a fully sealed filter, remember that axial fans suck at sucking. Your axial box fan will actually draw air around the front corners and into the fan, even without a filter on the back. And with how easy it is to reduce the fan speed of a box fan, you're better off allowing some air through so that the fan runs at a more efficient and safer speed.
5) A filter on a box fan is definitely better than nothing and a good, cheap short term solution. RAP units are great for long term use and capturing all sizes of particulate. I'm not going to try to sell you my brand, but there is one product I advise you avoid. Recently, Lasko has released a box fan with a filter slot as a 2 in 1 product. I've tested this thing with the provided filter and it does not perform nearly as well as a DIY filter you could make. Also, the filter slot does not fit standard 20x20x1 filters. The slot is designed for a slightly smaller, 19x19x1 sized filter, meaning that if you want to buy a replacement filter or a higher quality one than provided, you won't be able to. A normal 20x20 filter (which is actually slightly smaller in W x L nominally) can barely be squeezed into this fan's slot. Just buy a better filter and some tape instead.
6) If you look for DIY box fan filters, you will find examples with 4-5 filters in a cube shape. 5 filters improves filtration efficiency by ~50%, but costs x5 as much. That's $100-$150 in filters that will last just about as long as one filter while only cleaning the air 50% better. Just invest in a room air purifier unit of the same cost instead and enjoy much cleaner air over an entire year.