Cryptography won't help if you want a secret ballot.
The manager rounds up the workers and "suggests" they all compare how they voted. Those who voted "pro-company" will likely reveal their vote. So for everyone who refuses to reveal their vote, the manager checks a box next to their name on the attendance sheet.
There may be a voting system that provides all the desired guarantees, but I doubt such a system is also easily understood and trusted by the average voter.