I should know the answer to this:

Check is made payable to:

John Doe
Jane Doe

No and/or.

Can either one endorse and negotiate? Can Jane deposit into an account with her name only?

Assuming that your state has the standard version of UCC Section 3-110, it could be endorsed and negotiated/deposited by either one of them:

(d) If an instrument is payable to two or more persons alternatively, it is payable to any of them and may be negotiated, discharged, or enforced by any or all of them in possession of the instrument. If an instrument is payable to two or more persons not alternatively, it is payable to all of them and may be negotiated, discharged, or enforced only by all of them. If an instrument payable to two or more persons is ambiguous as to whether it is payable to the persons alternatively, the instrument is payable to the persons alternatively.