I'll answer chronologically.

In regards to what he could have done to prevent this:

The best he could do was actually never invade at all. The Romans lacked any crucial information on the horse archers, because they'd never faced them before.

By flight it was impossible to escape them,and as impossible to overtake them when they fled, and they had a new andstrange sort of darts, as swift as sight, for they pierced whatever theymet with, before you could see who threw them; their men-at-arms were soprovided that their weapons would cut through anything, and their armourgive way to nothing. All which when the soldiers heard their hearts failedthem; for till now they thought there was no difference between the Parthiansand the Armenians or Cappadocians - Plutarch

Therefore, whenever a historian gives an opinion on how Carrhae could've been won, you have to keep in mind that they have way more information on how horse-archers worked and what their weaknesses were.

Crassus had previously advanced on Parthia and took them completely by surprise. The Parthians did not have a standing army and didn't have the time to call upon one, due to the surprise. This left Ctesiphon and Seleucia, two vital cities to the Parthians, easy prey. Unfortunately, he turned back to Syria. This was Crassus's chance to win, but he threw it away.

Why did he take so many infantry to the open desert? Well, the Romans had fought on desert lands in Egypt and other places, so it wasn't completely foolhardy. Granted, fatigue probably played a part in his failure, but assuming Caesar marched on Parthia, he would be walking through the open desert too.

Crassus didn't build a fort for two reasons. First, if he spread out his troops to forage they would be easy pickings for the Parthians. Secondly, Crassus couldn't just settle in. That would leave his supply lines vulnerable and the Parthians would starve him out.

There's no real reason specified (in Plutarch's Crassus, at least), on why Crassus declined. There's only this:

But Crassus was in no small degree encouraged by Artabazesthe king of the Armenians, who came to the camp with six thousandhorsemen. These were said to be the guards and attendants of the king;and he promised ten thousand men clothed in mail and thirty thousandinfantry, who were to be maintained at his own cost. He attempted topersuade Crassus to invade Parthia through Armenia; for, he said, thearmy would not only have abundance of provision in its march throughthe country by reason of him supplying them, but would also advancesafely, having in their front many mountains and continuous hills, andground unfavourable for cavalry, in which alone lay the strength ofthe Parthians. Crassus was well enough satisfied with the zeal of theking and the splendour of the proffered aid; but he said he wouldmarch through Mesopotamia, where he had left many brave Romans; uponthis the Armenian went away.

Crassus definitely declined his officers' advice. This was because he wanted to win a victory quickly, to at least put himself on the same level as Caesar. The officers gave up, eventually.

Most historians' agree that as soon as Crassus fell for Ariamnes trap, he'd lost. No matter what he did, he wouldn't win. At most, he would retreat with a shattered army.

Now, onto your title question. Yes, the hollow square has been used. Napoleon used it to great effect in the Battle of the Pyramids, and it was definitely effective against cavalry. I won't comment on why, mainly because I'm not that well informed on pre-modern warfare. Also, even if the Romans were to be enveloped, it wouldn't have much of an effect. In a (static) hollow square, each flank faces it's side. Therefore, if the Parthians charged their flanks, it would be the equivalent of a frontal charge.

Source: Plutarch's Lives, Vol III, Crassus