There is nothing that should "blow up" the PIC, that would be a very unusual situation. In my 15+ years of using PICs I have only blown 2 up that I can remember, and both times was from connecting a high-current +12v supply to the Vdd pin. I have also killed a couple of ADC pins on PICs, due to overvoltage etc but that never stopped the PIC from being programmable.
If something is killing your PIC so it cannot be programmed you must be doing something seriously wrong!
You should be aware that the PIC16F628 and 16F628A have a LVP pin, so your programmer needs to handle the LVP pin properly to program the PIC, which means the LVP pin RB4 must be kept LOW (a 1k resistor to 0v) to do the programming. If for some reason your programmer has programmed the CONFIG to be _LVP_ON then you need a programmer that pulls the LVP pin low as required. You did not say what programmer you were using?
If the cap is charged up to a high voltage and you connect it to the cap meter then there is a possibility it can damage the PIC RA0 pin. But the voltage would need to be > 5v and have enough energy (a large enough cap) to blow the PIC pin protection diode on RA0! That is not that easy, the pin protect diodes are good for a couple hundred mA peak, so you must be doing something pretty bad there!
If you really think you might be connecting large charged-up caps to the meter(!) then you can add a resistor in series with the RA0 pin, this will limit peak currents into the PIC RA0 pin protection diode.
However you said "... testing a couple of 1 mfd, a 2 and a 6 mfd film cap that had been used since the summer in motor tests that the problem surfaced." I really doubt that you blew the PIC by connecting a 6uF cap that "was used last summer".
The RA0 pin is rugged, the PIC comparators are rugged and proven, the most likely thing is a PIC latchup reset issue. Possibly with a CONFIG issue that has programmed the PIC with _MCLR_ON (which explains the PIC reset issue) and LVP_ON, (which explains the unable to program issue).
Like I said my meter gets used all the time and I don't baby it, it has had caps connected that have been pulled from the breadboard and may have been charged to 12v or so, and the cap meter has not played up in any way not even as much as a single latch-up. Please check your programmer and CONFIG settings.