How many zeros must be in a bitcoin share for a pool to accept it?
First of all, Bitcoin does not actually use the number of zeroes for the difficulty. Rather it interprets the hash as a number and checks that the block hash is less than or equal to a particular target. This makes the hash have several leading zeroes, but those zeroes are not being considered directly.

The target for a share varies by pool, and usually, by miner as well. Miners can choose a target that matches their hash rate so that they aren't submitting shares too quickly or too slowly. Of course the shares are weighted by difficulty (i.e. target). So a share with a higher difficulty (lower target) is worth more than a share with a lower difficulty (higher target).

Most pools offer setting the difficulty to many values. For example, Slushpool lets you set it anywhere between 128 and 500,000.