I admit having a little difficulty--a personal problem due to irrevocably lost gray matter--following the example in the question. I got lost right away because two expected 1st cousins are mentioned at the start--neither of whom know their biological fathers--but I don't see where that fits in. The rest of the question describes cousins B and C whose results show an autosomal DNA sharing of 60 cM. Which means B and C can't be 1st cousins; can't even be 2nd cousins. Using Shared cM Project findings, not lower theoretical averages, the most likely relationship between B and C would be 3rd cousins or half-2nd cousins 1x removed...which would put the MRCA generationally back at the 2g-grandparent level, so not aunt A's parents or even her grandparents.
Either the 60 cM is a single segment among a total of about 600 to 900 cM sharing, or B and C aren't 1st cousins. Assuming it's a single segment, B's half-brothers (and here my mental sketch is that B and the brothers share the same mother, but that B has one father and the boys another) have been tested. Their results compared to aunt A's would immediately tell us whether aunt A is the sister of B's mother or her father. If aunt A is the sister of B and the brothers' mother, all of the siblings should match aunt A in the range of about 1,400 to 2,100 cM, but not match the brothers at all if aunt A was sister to B's father.
If we are at the level of 3rd cousins, not 1st, it's important to remember that meiosis and combination mean that the percentages of inheritance described as mathematical averages are only that: estimated averages. That high degree of variability is necessary for genetic diversity and survival of the species. And that's why the Shared cM Project shows that documented relationships as close as 2nd cousin 1x removed and 1st cousin 3x removed can show zero autosomal DNA sharing. And why work by Brenna Henn (former lead scientist for 23andMe) et al., shows that over 10% of our 3rd cousins will share no detectable DNA at all with us, that 54% of 4th cousins will be undetectable, and at the level of 5th cousins, 85% are DNA invisible.
Some additional detail--anonymous, of course--might help us provide more useful input.