On January 20th, 2005, a giant sunspot named "NOAA 720" exploded. The blast sparked an X-class solar flare, the most powerful kind, and hurled a billion-ton cloud of electrified gas (a "coronal mass ejection") into space. Solar protons accelerated to nearly light speed by the explosion reached the Earth-Moon system minutes after the flare--the beginning of a days-long "proton storm."
A Hydrogen atom is a proton with an electron, so in a solar flare or mass ejection you do get a lot of protons (ionized hydrogen) potentially hitting the earth. You also have a lot of electrons ejected and overall the solar flare is charge neutral. You can also get cosmic rays (not from the sun)- a lot are protons at very very high velocities.
And yes, in the upper atmosphere you have a lot of very reactive atomic oxygen, ozone etc.
However, I think with the magnetic field protecting the earth is one factor to consider, if you look at the Van Allen belts you will see the inner one is dominated by electrons, and the another farther from the earth is protons and ions. So that helps keep the protons away from the upper atmosphere. But for a big event, if protons are entering the atmosphere they would first become atomic hydrogen when they got an electron and would react with what could be available, but in the upper atmosphere they could be come ionized again. 473
Same with the atomic oxygen in the upper atmosphere low earth orbit, both would be bombarded by UV from the sun, other energetic particles etc. So it can be somewhat complicated. Through diffusion, perhaps some small number of the protons from the sun might ultimately make it down deeper in the atmosphere, and some might get kicked out into space by another energetic particle.
The other issue with making it rain, is that you need to have some amount of humidity, and that becomes hard to define in the upper atmosphere, you also usually want to have a nucleating particle for the rain drop to form around. In a cloud chamber you can see the path of a charge particle as it helps nucleate droplets, but that is becasue of the very high humidity and being near saturation. So if you had a bunch of the solar protons hitting an area of high humidity near saturation perhaps it could help seed rain, but since the magnetic field is protecting the earth, usually the particles that get closest to the ground are near the poles, and it is pretty dry near the poles since it colder.
So I would guess it would be hard for a solar storm to kick off a rain storm, and the amount of hydrogen that really gets into the atmosphere is also pretty small too. I don't know of any correlation between solar storms and rain.... so I would guess no.
If you are interested in solar flares etc. The NOAA Space weather page is here 473