Can a job force you to give up rights given to you in the Constitution?
Well, we know that it CAN,
by virtue of the fact that it already HAS happened,
but I believe that your question implies the issue
of whether u 'd be judicially vindicated in litigation,
if and when u challenged the practice.
As an example, a friend of mine applied for a job
as a bailiff in the NY judicial system. He was required
to surrender his guns, for which he had a license,
for several weeks, until he completed his training.
Despite the 2nd Amendment, during that time, he remained un
such that he cud not defend himself, nor his wife nor his child
from the violence of man or beast. Being un
means that u just take your chances
with getting killed.
My friend wanted that job enuf to risk his life
and to risk the lives of his family. He did not sue.
On the other hand, if an atheistic employer prohibited
an employee from going to Church on his own time,
most likely the employee 'd win a judgment in court,
vindicating his private rights, or if a Democrat employer
banned Republicans from voting at election time,
on pain of loss of their jobs, most likely this
wud not be judicially tolerated in litigation.
There r many other possible examples.