Ultrashot, a way of testing out whether it is due to bad breathing or just general unfitness is to generally have two stages next time you're out running.
The first stage, would be a medium paced run, with the breathing pattern established at a comfortable pace. Continue that for a few miles/minutes, whichever you have time for.
The second stage of the test can be slightly difficult to gauge. Mainly, this involves alternating short sprints with a slightly-slower-than-the-meduim-pace run earlier. Therefore, I guess you could do something like 50 metres *near* max sprint then drop down to normal or slightly slower pace after that for about a minute, and then repeat. Generally, if you can only do about two or three of these, you're generally not quite there, fitness wise and will need to be more patient before gradually building speed into your running.
This will allow you to assess whether it is bad breathing pattern (as in the first stage, if you have bad breathing pattern, you'll be out of breath and feel pain everywhere in your body.....generally) or lack of fitness for running (in the second stage, as explained).
However, from reading the line of:
"...pushing on the spot where it hurts kind of really hard and breath out as much as possible..."
Actually sounds more of a cramp issue in your, I presume, calves. Breathing in as much and as out as possible (but not hyperventilating!) does increase the oxygen flow to the muscles, which will help the muscles to break down teh lactic acid. I presume that the muscles are constricting on their own or seem to have a mind of its own? IF that is the case, streching will also help to do before hand.
I must confess that I don't run, but I do a lot of swimming. When I have had a holiday from the swimming training and have come back to train again, I generally have to start slow and the build up, or I'll just get loads of cramps and lots and lots of pain....in my lungs and body, cos of the need to push so hard for times and staminia. Thankfully, I don't do so much of the swimming..
I hope this helps
Submerged