How can I deionize water to prevent brown eye stains on my dog?

The iron in your water is present in two forms. The majority of it is iron (III) oxide, also known most commonly as rust. The second is ionized iron, which is Fe²⁺.

Iron oxide is insoluble in water, and if enough of it is present (usually happens in well water) then the water will start to have a brown/orange tinge. It is not filtered by Brita filters. They use activated carbon filters which will not filter out iron.

Reference for this. See section called 'What Activated Carbon doesn't do'

Fe²⁺ is usually not present in in high quantities in water, as it is a rare form of iron, and in drinking water it is not stable and precipitates into iron (III) hydroxide, which is also insoluble and rust colored. Reference. There still some remaining though.

To fully filter out the ionized iron and the solid iron, you need a system that oxidizes the ionized iron and a filter (sediment filters are fine) to get the particles that become oxidized or were already present. More information here.

This is all very complicated and can be expensive though (unless you just want to buy a sediment filter, which can be 10-15 dollars).

A less expensive option would be to buy a large jug of water at a grocery store that was treated by reverse osmosis, as this removes most of the iron. Make sure you check out the standards of the brand you are buying though, because some filter out more things than others.