Hi!
This takes me back.

I remember when they first came out some thought they were "unbeatable", but several games teaches you where they are vulnerable. Some tips:
1. Do not engage in long range fire fights unless you have built such a long range force optimally. If your not playing Eldar, this strategy of exchanging long range pot shots will probably lose you the game against Squats. Note that in this type of game you basically giving up on maneuver in favor of raw firepower. This is the hardest way to defeat them and requires a tight focus to achieve. When you go for the shooting solution remember to focus fire. Do NOT split fire since the squats break point is too good for halfhearted measures! Shoot infantry type targets in preference to all else. They are soft targets (low armor saves if any) and are the key to route the squat player since their loss gives the highest VP's and limits their objective taking.
If your going for their super heavies with this tactic, its an ALL OR NOTHING affair. Usually you need to open the assault with something that takes down shields quick like Eldar pulsar blasts, ork gargant ball round or warp/vortex missiles. GO BIG or go home with your assault against them, nothing short of these things will take them out quickly enough.
Note that basically the Eldar and then the IG as a distant second can pull off these "shoot 'em up" kind of games. Its hard for chaos and nearly impossible for orks and tyranids.
I suggest letting this strategy for when you are VERY experienced dealing against squats. It the hardest to pull off.
2. Engage quickly and often. The easiest way to beat the squat is what I call the disruption strategy. A proficient squat player will form a well positioned "baseline" where his precious infantry and other assets can mutually support each other and be able to lay a curtain of withering fire only the most long ranged armies can deal with (see number one above).
Therefore the best way to beat them is to disrupt that base line. The easiest way is using fast attack armies like space marines. Use drop pods and thunderhawk gunship and even teleporting terminators. The idea is to engage as much of the squat forces (go for the infantry mostly) and tie them up. The reasoning is that engage squats can't shoot elsewhere but at the most immediate threat. Since their number is small they are very easy to overwhelm. A tactical company in drop pods or assault marines deploying from the air via thunderhawks can totally tie up a whole squat army.
Do NOT worry about casualties. Every unit you lose is worth tying up that much squat firepower. More importantly the rest of your army can advance unhindered since the squat force it tied up defending from this disruption attack.
Throw in your own bike or landspeeder fast attack behind this and the squats will be VERY hard pressed.
Imperial titans, of course are the easiest to equip to deal with squat praetorians, especially warlords with their 4 weapon hard points.
This disruption tactic can be done with tyranids via mycetic spores, ork using that lander and chaos marines using their own t-hawks and IG using moles can accomplish the same disruption.
One particularly nasty trick would be to land via t-hawk a full company of Death Guard chaos space marines and unleash their legion power amongst squat infantry and watch the VP's accrue as the plague kills them off!
This is the EASIEST method to beat the squats. I have achieved one turn knockouts with less experienced squat players and even experienced ones can find it quite difficult to defend against this.
Most other strategies are permutations of these two, some with more than a little of one or the other.
Since you played Eldar the key is focus their fire. One ploy I like is using an Eldar titan with double pulsars. A potential of 2d6 shots is HUGE. ON average you are guaranteed to strip even the mightiest squat mobile fortress of ALL its shields. Use terrain to hide the titan and their great maneuverability and speed can get their in position and fire the pulsars which have a range of 100cm! Follow up this strike with tempest tank strikes and their high armor save modifier should punch some good wholes in the squat praetorian. Prism cannons in tandem can make an unshielded squat praetorians day pretty sour....
Use wave serpents to bring in elite striking scorpions to confront the squat infantry. The serpents shield is virtually impregnable and can even use it offensive before disgorging its lethal cargo.
Use doomweavers to plaster those buildings and other terrain the squat player uses to hide his precious infantry. A turn squat infantry moves to avoid the webs is a turn they can first fire!
Jetbike companies! A swarm of these guys can not only blunt the squat bikes, but are enough to wipe them out and keep going! Use two such companies and the squats are in for a short day...
Want to laugh? Bring the avatar and have it charge straight for the squat super heavy. That fixed 2+ save can do wonders.....
Its all permutations of strategies one and two. Focus your fire with the right units and disrupt, disrupt, disrupt! A squat player that is permitted to keep an intact baseline will cut you to pieces! Do let his baseline go unchallenged!
I can field any directed questions on the matter, good luck!
Primarch