The only way i can imagine pushing is to place your shield on the soldiers back whilst a guy behind you does the same to you. This seems like it would be bad for your balance, hurt you, maybe even make you fall over. I can't think of anyway to make sense of this during charging.
Mostly because that's generally not how premodern armies approached infantry combat; you kept your shield between you and the enemy, relied on the men next to you to cover you from attacks from the side, and thrust with your sword or spear. For hoplites, there's no point in having a spear if you're going to be fighting within inches of each other. Greek hoplite warfare generally involved quite low casualties; if entire phalanxes were bowled over as a result of this alleged shoving match, the slaughter would be absolutely appalling.
The idea of the "shoving match" is increasingly regarded as false, and rightly so, in my opinion. It is simply nonsensical.
Just to walk you through why the shoving match idea of warfare is one that's increasingly considered false (i.e. why nobody would want to do it):
It's very easy to counter. If you start pushing with all your might against the enemy line, it's very easy for them to just stab you. After all, you're putting all your attention and energy into pushing, so how are you going to stop several rows of people from skewering you?
It doesn't serve much of a point. What are you going to gain by pushing the enemy? Move them back a few feet?
It's incredibly exhaustive. Pushing against another line of soldiers is going to wear you out pretty quickly if you really are giving it your all.