I don't think your code will move them in a circle. It might spawn them in a circle, although the code for that looks a little weird for that too.
Let's go through you update() method in the hastebin link.
The first important thing is -- where is this update code? If you want to move the particles it would need to be in the particle class, but I kind of suspect you have it somewhere else like in an entity.
So your first mistake is that your code isn't moving the particles, it is just spawning them constantly. It might also be possible to used particles in fixed positions and "animate" them by killing and spawning them quickly but I don't think it would work well visually. So I think you need your particle class to remember the center of the circle and the angle, and then in the update method for the particle you would add some angle and adjust the position (not respawn) accordingly.
Additionally, I think your math is still suspicious. Your code to use trigonometry to adjust the X and Z positions looks like: double angle = Math.toRadians(degrees);
double vx = (pos.getX() + 0.3) * Math.cos(angle) - pos.getZ() * Math.sin(angle);
double vz = pos.getX() * Math.sin(angle) + pos.getZ() * Math.cos(angle);
I'm pretty sure the math is wrong here. The X and Z positions are the values in the world, so they can be numbers like 600 or 0 or -600. Instead I think you should be multiplying by a radius for the circle as the vx and vz fields presumably indicate the difference in position relative to the center. Also, you're mixing up the two dimensions. Sin and Cos functions already take and angle and separate it into the two dimensional components, so I think it should be something more like:
double angle = Math.toRadians(degrees);
double radius = 2.0D;
double vx = radius * Math.cos(angle);
double vz = radius * Math.sin(angle);
Then instead spawning the particles again, you should just move them. Also, you shouldn't move them based on current position, but recalculate from the center of the circle. So the new positions would be something like (this is pseudo code because you need to put the code into the right place with a field for the center:
this.setPos(particleCenterX + vx, this.getPosY(), particleCenterZ + vz);
Summary
Putting it all together:
The movement code needs to be in the update for the particle class which runs on the client side. You can't just keep spawning particles.
I would have the particle "remember" where the center of the circle should be.
You need to have a sense of "radius" for the circle.
You simply update the particle position with very simple trigonometry such as center + radius * cos(angle)