Does gas chromatography follow the "like interacts with like" rule of thumb?
For GC, the most significant factor affecting separation resolution is differences in vapor pressures, boiling points if you will, between analytes.  While column phase selection, oven temperature programming, detector type and temperature, and other factors will have an effect on the efficiency of separation, IMFs are not generally important for the majority of separations.  The "like interacts with like" principle does apply to GC such that in the case you describe, nonpolar analytes will spend more time interacting with the stationary phase while polar will have little to nearly zero interaction and elute first.  The separation of the various nonpolar compounds will, holding all the other column conditions constant, "boil down" to varying vapor pressures.

Edited for clarification:

To begin, I think it beneficial to keep in mind that the question you pose is sufficiently clear we are dealing with polar analytes using a non-polar GC column.  Regarding proprietary column selection, temperature programming, and the choice of various other method parameters, these things are necessarily important in an efficient and successful separation (generally called method development) but they do not do much to answer the principle questions asked about whether 1) polar compounds are less retained on a non-polar column, and 2) does "like with like" apply to GC.  To that end, I will again address these two primary questions and then try to clarify another issue brought into the discussion that does not bear directly on answering the questions as asked.

The most direct answer to your first question is simply, yes, polar analytes do have shorter retention times than non-polar compounds when using a non-polar column because polar compounds have little interaction (partitioning) with the column's stationary phase.  The polar molecules of interest have little affinity for the non-polar stationary phase, i.e. hydrophobicity.  Had it been a polar stationary phase column then IMFs would be significant.  You mention making things comparable by considering compounds of similar boiling points which is precisely why a ramping temperature program is most beneficial for efficiently resolving such a separation.  This too, however, is a method development consideration and thus an example of the applied science of gas chromatography taking advantage of pure science principles.

Considering your second question, this, too, is most directly answered with another yes - the "like with like" concept does apply here, but as you point out, this is a rule of thumb, and as such, a conceptual aid not strict principle.  The qualification about hydrophobic effects in the stationary phase you make in your wording of the question actually explains, as well as I could, what is happening.  Also, as @Blaise points out, the statement with which you end your original question referencing a previous user (I [remember] reading that the interaction between a polar molecule and non-polar molecule is stronger than the interaction between two non-polar molecules.), is in error: the interaction between polar and non-polar molecules is not stronger than between two non-polar molecules.  Perhaps what this person meant is the interaction between two polar molecules is stronger, and relatively much stronger, than interactions between two non-polars.

With regard to your request I clarify why IMFs are not generally important (in this case), it must be remembered that the analytes in GC are not in "pure liquid form" but, rather, in the gas phase and they are completely surrounded by inert carrier gas molecules - little chance for IMFs to be of significance.  The polar molecules to be resolved do not spend a significant time partitioning into the stationary phase so they cannot form immiscible droplets as you brought up.

With respect to @SteffX, your question is about GC specifically and is not a question of what is the best separation method to use.  I take your question to be about using GC (on small sample sizes, right or wrong on my part) to effect a separation for analytical purposes, not as a preparatory separation like distillation.  Also, your original question(s) are with regard to gas chromatography in general and make no reference to column packing, bore, or length (other than being of a non-polar stationary phase) and I tried to answer in general terms.  However, since capillary GC columns were brought into the discussion, I think it deserves clarifying that, in fact, most capillary column separations are not isothermal.  Without going too deeply into something you were not asking about, it should be noted that isothermal temperature separations are a simpler and therefore preferred method development as long as you have good analyte resolution; the more common practice is to program a temperature gradient to resolve similar boiling point compounds - part of your original Question 1.  Lastly, Van der Waals force(s), a subject for much definitional discussion in its own right, have little importance to your polar analytes at GC conditions, and are moot almost before the sample hits the injector.

I hope this has been helpful and, in my opinion, I think you already have a good understanding of this.