When integers join the queue

Python 2, 56 53 50 bytes

q=[]
for i in input():q=[[i]+q,q[:i]][i<0]
print q

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Dequeue is -1. This trick allows easy pythonic slicing of the queue.

Mathematica, 102 bytes

Definitely not the shortest solution, but I couldn't resist because it's kind of perverse.

r=Reverse@{##}&
a_~f~b___:=b
f[a_,b___,]:=b
ToExpression[{"r[","f["~Table~StringCount[#,"]"],#}<>"]"]&

After some helper functions, this defines a pure function that takes a string as input: in the string, numbers are separated by commas (whitespace is optional); the dequeue character is "]"; and the list does not have delimiters in the front or back. For instance, the first example in the OP would be input as the string "45,],],37,20,],97,],85". The output of the function is a list of numbers.

The function counts how many dequeues "]" are in the input string, appends that many copies of "f[" to the front of the string, and then surrounds the whole thing by "r[...]". In the example above, this produces "r[f[f[f[f[45,],],37,20,],97,],85]"; notice the brackets are balanced.

Then, ToExpression interprets the resulting string as a piece of Mathematica code and executes it. The function f is conveniently defined to retain all its arguments except the first (and also ignores trailing commas; this is necessary to handle dequeueing empty queues anyway), and r converts the resulting sequence of numbers into a list of numbers in the right order.

Retina, 30 bytes

1+`\d+,(.*?)X,?|^X,
$1
O^$`\d+

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Repeatedly removes the first number that's (not necessarily immediately) followed by an X together with that X, or an X at the beginning of the string. Then reverses the remaining numbers.