Can I measure entropy through compression only? Say I have a file containing words from the English language.  Not meaningful text, just a repetition of a few words.  The original size is 1559454 bytes and it is compressed by a certain program to 75742 bytes.  (A very good compressor, it seems.)  Can I estimate the entropy of the source producing such file using essentially the byte reduction?
Can I estimate the entropy of the source producing such file using essentially the byte reduction (when compressing the source's output)?

No, at least for Shannon entropy. Entropy is a characteristic of the source, that can't be determined from a finite sample of the output of the source. It's can't even be estimated in practice without information about what the source is, and the question gives no such information.
The best we can is estimate a plausible upper bound of the entropy of the source that generated the file: $75742/1559454$ entropy bit per output bit. That's not a mathematical certainty: it's mathematically possible that a perfectly random source generated that file (the probability is $2^{-8\times1559454}&gt;0$). And that's not a practical certainty either, unless we add a vague hypothesis that the observed characteristic goes on: a device which output consists of the $1559454$ fixed bytes followed by indefinitely many uniformly random bytes is a source with $1$ entropy bit per output bit.
We do not get any lower bound better than $0$. Proof: any deterministic program that indefinitely outputs is a source with zero entropy. And it is trivial to make a deterministic program that indefinitely outputs a sequence starting with the $1559454$ bytes (perhaps, repeated). That's not merely theoretical:

The program which output consists of the repeated sequence of two bytes a  produces output which first $1559454$ bytes matches the problem statement, even for some actual compressor; say, one that never compresses by more than a factor of 20 (that's quite plausible for e.g. an audio compressor).
Even if we restrict to files that could reasonably occur in actual use, it's entirely possible that a program designed to obfuscate a file into english words, written as an (elementary and rather poor) steganography tool, has output that matches the problem statement (for an actual text compressor) when fed as input a file of about $50000$ bytes, including if it is all-zero.


Conclusion: practical procedures that attempt to assess the entropy of a source from its output only are doomed to, at best, give a plausible upper bound of that entropy (and can do so only under the hypothesis that the output tested is representative). Computing a compression ratio is one such method.