I don't know your breed but in mine colors like Squirrel were a DQ. But that is from breeding a chin with an agouti color like fawn, chestnut etc. In my breed chin bred with a self (as black, tort or blue would be) was OK. Chin would be dominant so you'd know which ones should not be bred to an agouti of any kind.
Chinchilla *is* an agouti color; it's basically chestnut with a gene that "turns off" production of yellow pigment, so the orange/fawn band appears pearly/white instead. It is written
<A_ B_ cchd_ D_ E_>.
Squirrel, which in mini lops is an accepted color called blue chinchilla, is just a dilute chinchilla
<A_ B_ cchd_ dd E_>
so breeding a chinchilla with a fawn (dilute red <A_ B_ C_ dd E_>) might produce squirrel if the chinchilla also carried a dilute gene. But breeding a chinchilla with a chestnut would not (unless both chin and chestnut happened to carry dilute).
As eco2pia noted, breeding a chinchilla to a black (self) would likely produce self chins
<aa B_ cchd_ D_ E_>
down the road, which can wreak havoc on a show breeding project.
I have struggled with this for generations in my black Satins, many of which are genetically self chins (and some of my blues are self squirrels). These bunnies are black due to the homozygous self genes, but the chin shows through in their blue-gray or marbled eyes. Adding more confusion to the mix, their eyes often change to brown as they mature, so you really cannot tell which are black and which are self chin. No problem on the show table, but it's a bummer when you breed your "black" to a REW and end up with a whole litter of lousy chinchillas and squirrels! (They tend to be lousy because you've been breeding chinchillas all along, but you haven't been able to see the details of ring color etc and so have not been selecting for good color.) Since chin is recessive to full-color C, and is hidden under self, it can lurk in many generations of selfs (9 and counting in my herd!).
Incidentally, breeding a chin with a non-extension color like tort is even worse from a show perspective, because you'll eventually get ermine (non-extension chin), which as far as I know is not an accepted color in any current ARBA-recognized breed. (The Czech Frosty is an ermine but that's still in progress.)