Hassidim generally wear three types of shirts:
A regular white dress shirt.
As above, but with reversed buttoning, i.e. right-over-left. Pocket remains on left side of the chest. Basically identical to women's dress shirts, save for the bust shape.
A "rebbishe" shirt with three or four differences from the above: 1. The sleeves have no cuff, but instead widen until about twice normal size at the wrist, where it has a simple stitch like at the bottom of the shirt. 2. The buttonable opening ends on the belly, where the button and hole parts are sown together in an upside down triangle (▽). 3. There are no pockets on the chest. 4. (Optional) The shirt may be very long, almost to the knees.
Each type is available from several different brands, with negligible variations. All have standard point collars. Many hassidim will close the topmost button, even as they are not wearing a tie, thereby looking similar to a clerical collar.* This might be why you think the collar is non-standard.
As HodofHod correctly stated, The right-over-left buttoning is to put chesed over gevurah.
Source: Personal experience. (I wear number 3, including 3.4.)
* Therefore, some hassidim specifically leave the topmost button open – to avoid looking like a Christian cleric.