Not at all. The bit about white gills is probably referring to Amanitaceae, which include some marvelously toxic species that destroy the human liver. But the advice is way too general to be useful. Brown-gilled species are not universally edible, and white-gilled species are not universally poisonous.

To forage safely for plants or mushrooms, you should know the identifying characteristics of your target species and any non-edible lookalikes.

There are definitely no general rules for determining mushroom edibility.

A great resource is Mushrooms Demystified by David Arora.

I did a couple minutes of looking to find a poisonous mushroom with brown gills:

https://en.wikipedia.org/wiki/Galerina_marginata

Such simplistic rules are a bad idea when foraging. Before you eat anything, positively identify it.

There are no simple rules like that which can reliably differentiate edible mushrooms from poisonous ones. The best way to start foraging is by learning some relatively unmistakeable edible species and only picking those.

No. The deadly webcap is an example of a poisonous mushroom that had brown gills. It can be found in northern Europe. Further, there is a broad category of difficult to identify mushrooms referred to as "little brown mushrooms" that includes poisonous brown filled species.

The notion that brown gilled mushrooms are safe whereas white gilled mushrooms are not sounds like region-specific folk knowledge, and given the way mushroom species migrate and the existence of relatively rare mushrooms, I wouldn't count on it to be reliable anywhere in the world. Positive identification is the only safe way to select edible mushrooms, and as this article points out even that bears the risk of false identification, particularly if you aren't fully aware of what other similar species to check for.

Absolutely not