You said: "First, I have a 3rd cousin match from Ancestry who uploaded his info for me on gedmatch.com, and I see that he shares no X DNA with me. We're trying to figure out which of his great great grandparents we share"
Your cousin is male. Males only get one X and it's from their mother. The simple way to think of this is that the X cannot be passed down to two males in a row.
Therefore of his 16 great great grandparents, since you don't share X, you could be related through any of the 8 great great grandparents on his father's side, i.e.:
- his father's father's father's father
- his father's father's father's mother
- his father's father's mother's father
- his father's father's mother's mother
- his father's mother's father's father
- his father's mother's father's mother
- his father's mother's mother's father
- his father's mother's mother's mother
(I've bolded the "2 males in a row")
or through these great great grandparents on his mother's side
- his mother's father's father's father
- his mother's father's father's mother
- his mother's mother's father's father
The 5 great great grandparents you likely are not related to are:
- his mother's father's mother's father
- his mother's father's mother's mother
- his mother's mother's father's mother
- his mother's mother's mother's father
- his mother's mother's mother's mother
So X doesn't help you very much, as only 5 of the 16 candidate ancestors are not likely.
The reason why I say "likely" is that a woman gets the one X chromosome her father had, but only one of the two X chromosomes that her mother had. It is therefore possible that for one of the 5 great grandparents listed just above, that the common ancestor's segments did not get transmitted down mother to child. I'm not sure of the exact probability here, but at 3rd cousins, it may happen but not too often. This means none of the 16 great-grandparents can be assuredly omitted. So unfortunately, the bad news is that the X chromosome is likely not to be of too much help to you.
Instead, I would recommend you look at the Autosomal DNA and the people you and your 3rd cousin both match with on a particular segment. This requires chromosome matching, which you can do at gedmatch. The great great grandparent you are looking for may be able to be determined from the common ancestors of you, your 3rd cousin, and the people you both match with.
Then you asked: "Additionally, I have a cousin match on ged match who shares 77.3 cMs worth of X DNA with me and no autosomal DNA. How much of a cousin match is this?"
The X chromosome does not crossover as often as other chromosomes. Therefore a match on the X chromosome may continue many more generations than it does on the other chromosomes. So it is quite possible that a 5th or further cousin may share on the X but not on any of the other chromosomes.