How would an infinite stream of bits possibly encode an integer in the first place? All integers are finite, so unless you did something stupid like assign the infinite sequence 11111111... to the integer 37, an infinite number of bits should never correspond to any integer.
The idea is that you invent a system where each integer corresponds to a finite bit-string, but the lengths of these strings must be unbounded. Unary is one such code.
Then you could set up a computer program which decodes these strings, so you feed it one bit at a time, each time asking it 'has an integer been uniquely specified yet?' The OP's claim is that whatever encoding method you come up with, he can come up with an infinite string that will make your program keep saying "no integer has been uniquely defined yet" forever.
So, nobody is encoding integers as infinite strings, although there's no particular why you can't, in fact the overwhelming majority of possible encodings use infinite strings.
Followup to: What's a "natural number"?
While thinking about how to make machines understand the concept of "integers", I accidentally derived a tiny little math result that I haven't seen before. Not sure if it'll be helpful to anyone, but here goes:
You're allowed to invent an arbitrary scheme for encoding integers as strings of bits. Whatever encoding you invent, I can give you an infinite input stream of bits that will make your decoder hang and never give a definite answer like "yes, this is an integer with such-and-such value" or "no, this isn't a valid encoding of any integer".
To clarify, let's work through an example. Consider an unary encoding: 0 is 0, 1 is 10, 2 is 110, 3 is 1110, etc. In this case, if we feed the decoder an infinite sequence of 1's, it will remain forever undecided as to the integer's value. The result says we can find such pathological inputs for any other encoding system, not just unary.
The proof is obvious. (If it isn't obvious to you, work it out!) But it seems to strike at the heart of the issue why we can't naively explain to computers what a "standard integer" is, what a "terminating computation" is, etc. Namely, if you try to define an integer as some observable interface (get first bit, get last bit, get CRC, etc.), then you inevitably invite some "nonstandard integers" into your system.
This idea must be already well-known and have some standard name, any pointers would be welcome!