No, there are no squatter rights in California. It is illegal to inhabit an abandoned property, and whatever monies you put into it for improvements and/or utilities is definitely not considered rent.
There is a legal concept called "adverse possession" which after five continuous years might entitle you to become of the owner of the property, but the requirements are quite stringent and all of the elements must be met before you can acquire ownership of the property in this manner.
Only if you have paid the property taxes, been there 5 years and met other criteria, may you make a claim for adverse possession.
Disclaimer: You should discuss your situation with a local attorney, as the above statement is a general overview and does not take into account all of the factual considerations in your case. Nothing in this statement creates an attorney-client relationship.