Summary
While in Block Stun, press the controller to Toward and then press the Assist Button that corresponds to the Assist Character you want to jump in and perform their Cross-Over Counter. A Cross-Over Counter costs one Level of Hyper Meter to perform.
Counter Intuitive
Again, the Assist 1 Button will call Assist Character 1 to perform their Cross-Over Counter and the Assist 2 Button will call Assist Character 2 to perform their Cross-Over Counter. The move that is performed is directly tied to the Assist Type that was chosen. Whatever their Cross-Over Assist action is, that's what they will perform when they jump in as a Cross-Over Counter. And yes, that means if your Assist was Morrigan with the Dark Harmonizer, performing a Cross-Over Counter will have Morrigan jump in and give Hyper Meter to herself, leaving herself extremely open to just about any attack you can think of.
Again, because the character jumping in is performing their Cross-Over Assist action, what was said about Cross-Over Assists applies to most Cross-Over Counters: they come out slowly and they have very little invulnerability. Proper use of Cross-Over Counters to actually hit the opponent in the middle of their move is very difficult. Usually, the only way to get it to connect is to counter a Normal Move the very instant the opponent chains it into another Normal Move so that the second Normal Move whiffs and you come in and hit them. But even then they can cancel the whiffing Normal Move into Hyper Combo and beat up your Cross-Over Counter.
Strategy Corner
What this means is, essentially, Cross-Over Counters are almost useless being used as an actual counter. One of the other most common uses for a Cross-Over Counter actually turns out to be saving the life of a character that is about to be chipped to death without using your X-Factor. Right before they die from Chip Damage, if you perform a Cross-Over Counter to a character who has full health or enough health to take the Hyper Combo and still be able to fight effectively, you may want to do that so the character who was about to die can start regenerating health as an Assist Character so they can live to fight another day.
Powering Up Your Counters
However, there is one really good advantage of Cross-Over Counters: they are performing the Assist action as the Point Character, which now means they have a lot more freedom... namely that they are in a Special Move Attack State. That means, as with any normal Special Move, you can cancel the Special Move into a Hyper Combo at almost any point.
So if your Assist action allows you to cancel the Special Move the very first frame it starts, this pretty much allows you to counter any move with a Hyper Combo! A great example of this is choosing Deadpool's Gamma Assist, Trigger Happy. This move can be canceled into a Hyper Combo the very instant he starts the move so what you can actually do to opponents is, while they are performing a Chain Combo on you, perform a Cross-Over Counter into Deadpool's Gamma Assist and, before the Trigger Happy even does anything, cancel that into the 4th Wall Crisis, which is a Level 3 Hyper Combo that counters attacks! So the opponent performing their Chain Combo will end up getting countered by a Level 3 Hyper Combo!
You don't even need Counter Hyper Combos to pull off these tricks. Many Hyper Combos just hit fast enough that they will outright beat the opponent's attacks. And other Level 3's that have a lot of invincibility will definitely beat up the opponent's moves very effectively. Experiment with your team and see what you can come up with.
Invincible Cross-Over Counters
Some Cross-Over Counters have invincibility all the way through start-up and into their first attacking frame. These can be used to smack the opponent off of you even without canceling into a hyper (though they are still punishable if your opponent has time to block the attack, naturally). The list:
Ryu α (Shoryuken)
Morrigan α (Shadow Blade)
Chris α (Combination Punch)
Felicia α (Rolling Buckler)
Chun Li β (Tenshokyaku)
Tron Bonne β (Gustaff Fire)
Arthur α (Heavenly Slash)
Zero α (Ryuenjin)
Haggar α (Double Lariat)
Jill γ (Somersault Kick)
Wolverine α (Tornado Claw)
Captain America β (Stars & Stripes)
Deadpool α (Quick Work)
Thor β (Mighty Smash)
She Hulk γ (Somersault Kick)
Spider-Man γ (Spider Sting)
X-23 γ (Crescent Scythe)
These were tested by having the training dummy (Ryu) fire a Shinku Hadouken on a blocking and cornered opponent at point blank. After blocking the Shinku, the defender attempted to tag in his partner via Cross-Over Counter. If that particular counter assist beat out Shinku clean on entry, then it was put on this list. There's a chance that some of these may be only projectile/beam invincible and not "completely" invincible, or that they may be vulnerable for just a frame or two before hitting, so if you find something that disagrees with this list, feel free to correct it.
The rule of thumb seems to be that if it looks like a Shoryuken-style anti-air, then it's invincible through start-up when used as a Crossover Counter even if that attack doesn't normally have invincibility when used in other circumstances. There are also some other obvious ones (Haggar) as well as some not-so-obvious ones (Chris).
There are no counters that are invincible all the way through startup, active, and recovery frames.
Furthermore, while most Cross-Over Counters are vulnerable before their attacking frames begin, all of them have at least a little invincibility when they initially jump onscreen. You can cancel this mid-air jump-in with an invincible aerial hyper combo (if your character has one) to make for a reasonably safe entry and counterattack.