1. Learn to play the "air bass".
2. Sync to a really good recording of Chris.
3. Rock on.
MM
What MM said, plus some thoughts --
I'm a huge Geddy Lee fan, and while growing up I realized that some of Lee's style/sound derived from his Squire influence. So then I listened to Squire as much as I could. From that, I learned a lot --
The obvious stuff is that 1) Squire used a pick, rotos, and a Rick. But these can be deceiving. Ie, Squire's Rick had been sanded/shaved so many times it wouldn't sound exactly like a new Rick. I personally don't use a pick, but using my index fingernail (similar to Lee up until the early 90s), bright tone w/growl with some EQ can get you in the ballpark. Then there's Squire's post-1970s Tobias/bartolinis tone, but it sounds like you're not interested in that, so I'll move along.
Stuff with Squire that people often don't talk about is his groove/sense of rhythm. He approached it from a kinda quirky angle, which I love. And, he also seems to swing between the extremes of a stacatto or legato vibe. My recommendation, which some people would scoff at -- don't necessarily try to learn Chris Squire's parts, but learn to play along with Bruford.
Lastly, it's often overlooked that Squire has an incredible sense of harmony from his days as a child in the church choir. When breaking his parts down, keep his harmony background in mind.
Have fun!
Jay