TN1 Visa Eligibility

I am currently completing my 3rd year undergrad at an American university (for a BA) applying for a summer job. Am I eligible for a TN1 visa this summer?

If the job you find is in a TN categoruy and your bach degree is related to that job, then of course -- as a Cdn citizen -- you are entitled to TN.

Otherwise, no.

My degree is relevant, but I have not completed it yet (I'm in the process). Will that be a problem?

You need the degree first to get TN. But you don;t need the TN until you start working, so you can certainly look for work now, but you'll need degree (as shown on your transcript) before any border officer will give you a TN.