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That will depend on the wording of the company handbook.
In employment law, if you have no employment contract that specifically states that you can only be terminated for cause, you are an "at will" employee. That means that you can legally be terminated at any time, with or without cause or warnings.
So, for this handbook to actually alter that normal "at will" employment status, it would have to have very specific language making it an implied contract. It would have to remove all discretion from the employer. If the handbook makes it clear that it does not intend to alter the "at will" employment relationship, then it wouldn't amount to a quasi contract and there would not be any legal basis for a claim for damages.