Wouldn't (or would not) is the conditional form of won't (or will not).
Don't (or do not) is the negative form of do.
Essentially, the sentence
We wouldn't say "I'm having my friend".
has the same meaning as:
If we wanted to be grammatically correct, we would not say "I'm having my friend".
In this context, the first part of the statement (the "hypothetical" half) is left out because we tend to understand the conditional phrase even without the "if" part of the sentence. Based on the context of the sentence, we infer the part that is missing.
Using "don't" instead of "wouldn't" changes the meaning of the phrase, but not greatly. For example, you could say:
We don't say the phrase "I'm having my friend" in English by itself because it is grammatically wrong.
We don't say "I'm having my friend" unless we mean "I'm having my friend over for dinner" because it doesn't make sense otherwise.
There are other sentences you could make with that phrase—these are just examples. However, the important point to notice is that it changes the meaning from being a hypothetical situation to a factual statement. Also, "would" tends to have a polite or teaching suggestion tone to it, while "don't" is less polite. "Don't" can be harsh or simply factual based on the person's tone.
(This is based on my impressions of the two from English-speaking Canada.)