Will this PSU work with my PC?

It should fit fine.

Thanks for the response, will the connections all fit fine as well? The graphics card I'm using is a AMD Radeon HD 7850

Yes they will, it is all standard.

Ok thanks, just one more question. I was concerned about the form factor of my motherboard being uATX and the form factor of the PSU being internal ATX. Why does that not matter in this situation?

microATX is only the size of the board, just a smaller version of an ATX board with all the same connectors and less expansion slots.

I would also consider the computer case to be uATX since it only fits uATX form factor motherboards and only has four PCI/PCI-E slot covers.

The HP OEM 300 Watt power supply is 150mm W x 140mm D x 86mm H (5.9 W x 5.5 D x 3.4 H inches).

The Antec NEO ECO 620C is 5.9" W x 5.9" D x 3.4" H so it's 0.4" longer in depth which is the only physical PSU case dimension that is allowed to vary in the ATX power supply physical form factor spec.

So that means I'm A-OK, right?  

Yes it is OK since it is a mini tower.

If there's enough room for all of the cables to fit between the new power supply unit and optical drive(s) you should be fine.