How do you address any challenges with getting young soldiers involved and having pride in your unit?

Buy Beer they will come

Sir,
I joined the Army back in 1992. My duty station was at the National Training Center at Fort Irwin, CA. One of the things I noticed was that even though Soldiers didn't get along with the leadership in some aspects, they did have pride on what they did, pride on the mission. I'll be honest, no one likes to conduct unit runs under a 100 degree weather, but we did it, and we had lots of fun with it. At the end of rotations, we did BBQ's at our motor pool...everyone showed up, even those just returning from their last missions from out on the field. The crazy thing is that no one wanted to go home, hahaha! I loved that stuff!
Towards the end of my career, you are right, trying to get the Soldiers together is like pulling teeth. I hate to say it Sir, but some of that falls on the leadership. When I was filling in as a Platoon Sergeant, I was overwhelmed with work from meetings, NCOERs, Awards, FTXs, etc., you know how it is. But we also have to make the time to talk to the subordinates. My Platoon Sergeant actually did that, he brought in Soldiers to his office and talk to them one on one, on a relaxed setting. He actually took notes of Soldiers concerns which made them feel like he actually care in which he did. Now, I'm not saying to bring in the whole company into your office and talk to all your Soldiers. But maybe this is something you can do with your XO, Platoon Leaders, even your 1SG. Which bring me to another point. Unless you are doing it already, and I know we get busy, but we have to show our face. Soldiers like to see who the Company Commander is, not just from a picture on the wall, but the actual person. Unit functions are great because it gets us out of the uniform which makes us a little more approachable.
You're right Sir, times have changed. Is not like it used to be, but it can be done. Good luck Sir!

Sir, the more you try forcing us to do things, the less people will want to do anything with the unit. Morale starts at that 9-5. If soldiers don't have any morale at work, a unit Ball, cookout, or anything else of the nature aren't going to raise morale. In my experience, the main drain to morale is toxic senior NCOs. If you shield soldiers from toxic leadership it goes a long way.

Take away their Xboxes, Gameboys, and cell phones. Maybe they could learn to be sociable again.