Can my manager try to get me to quit?
He knows that if you quit, you will have difficulty collecting unemployment benefits whereas if he fires you, you will have no difficulty collecting unemployment benefits. That is most likely his strategy.
You are right to resist resigning. Simply keep doing your job to the very best of your ability. Write down when he calls you and so forth so that you have a record of what has transpired in case it becomes necessary to be able to recall your situation later. Do not let him intimidate you to quit until you find a better job.
However, no one enjoys being in a workplace where one is not wanted. So, it does make good sense to be checking the major job boards (Monster and CareerBuilder) as well as sites such as LinkedIn and Craigslist at least once a day and submitting your resume to any job that interests you.
Do not apply to other jobs using a company computer or on company time. He can use that kind of misuse of company resources/time as justification to fire you for "cause". That could affect how quickly you will be able to collect unemployment compensation.