What is a word for trying to avoid contact when you see someone, especially eye contact?

You're looking for a single word, right? Avoiding "eye contact" for any reason (shyness, hatred, no interest, contempt) is simply "avoiding eye contact". There is no single word to cover all possibilities.

I came across that FRIENDS episode again, and realised that the word was 'blow off'. Ross was upset that Chandler blew him off. Totally ignored him. Acted as if he wasn't interested in him.

blow sth/sb off-- to treat something or someone as if that thing or person were not important.

A term for ignoring someone intentionally that also starts with a B is blanking someone.

blank [somebody] [transitive] (British English, informal): to ignore somebody completely

• I saw her on the bus this morning, but she totally blanked me.

[OLD]

It has more BrE usage than AmE, so maybe not what was on Friends (Black Books covered it as well). Otherwise suits: You would pretend not to see them, even avert your eyes as if glancing past them or deliberately turn your head away so much that you hurt your neck. It comes down though the word blank as in blanc, white, nothing, giving a blank look when you don't recognize someone.

TFD also gives:

21. slang to ignore or be unresponsive towards (someone): the crowd blanked her for the first four numbers.

The word you're looking for might be "opsablepsia".

The inability to look someone in the eye, or not looking into another person's eyes, while speaking.

_{https://wordinfo.info/unit/302
https://www.encyclo.co.uk/meaning-of-opsablepsia}

Also, the link https://wordinfo.info/unit/302 lists a number of fears, phobias and conditions related to seeing, formed with the Greek word/suffix -blepsia meaning sight.