[Looking for Diogenes]

Could a private company ask you to renounce your religion as a term of continued employment?

Anti-discrimination laws are intended to prevent that form happening. But a employer should be able to tell his employees to leave their proselytizing out of the workplace. An employee is there to get work done, not convert sinners.

[jwalsh07]

So the first amendment does apply to private employers in some manner.

[Dog Gone]

Absent arguably unconstitutional legislation, there's no reason why a company couldn't require you to renounce all religion as a condition of further employment.

It's a private matter. Similarly, there is nothing in the constitution which would prevent a company from requiring all employees to be some weird sect.