The amount of autosomal DNA shared by 6th or 7th cousins is quite small. Some will share no detectable DNA. The latest data of the Shared cM Project shows that seventh cousins share between 0 and 57 cM, with an average of 13 cM.
The Y-DNA plays no role in this. Your female match has two X-chromosomes, and you have an X and a Y, but assuming this cousin is on your paternal side you would not expect to share any X or Y DNA with her.
Autosomal DNA inheritance is quite random at this level, due to the cumulative effect of genetic recombination. The segments you share, if any, might be quite small.
If you upload your DNA to Gedmatch, you can do a One-to-one comparison but lower the thresholds (to say 3 cM) to pick up small segments. Interpreting small shared segments has to be done with caution as you may share these segments by pure chance rather than common ancestry.
DNA is not conclusive with many distant matches. A non-match indicates either no genetic relationship, or distant relationship (likely 6th cousin or more distant). With small segment matches, this means there is either a distant genetic relationship, or matching segments occuring by chance. The DNA can provide you with a hint in this case, but it is up to the genealogist to verify any relationship by DNA testing other potential relatives on this line, as well as scrutinizing the paper-trail to identify any "non-paternal events" or the like. My advice would be to use the DNA as a piece in the puzzle here, but be careful not to over-interpret it.