In the episode where Squidward travels in time, right before he breaks through the floor in the weird dimension place, he doesn't have his clarinet. Then, when he breaks through the floor and lands in the time machine, he has his clarinet. Then, when he says "I even miss Spongebob!" and gets home and opens the door, he doesn't have it again! Weird, huh? YAK!
The other dimension was like a dream world. This means that he could very easily have had his clarinet while he was in it. Since everything else about the dimension was fabricated. then when he returns to reality it is naturally gone.