I have an animal companion which has just gained its 5th HD. On page 291 of Monster Manual 1, the Improving Monsters section says that an increase in Size also changes stats.
Do animal companions (e.g. from Druid or the Wild Cohort feat) increase in size based on their HD, and if so, do they also gain the stat adjustments from MM1 p291?
No, in D&D 3.5, animal companions do not grow in size due to bonus HD. It does not clearly state this in the core rules but is explicitly stated in a Skip Williams rules article on animals on Wizards's site.
Like many things, this was improved in Pathfinder, where specific size increases are defined per animal companion type explicitly in the rules.
A druid's animal companion is not a standard monster or animal. The bonus hit dice do not change its size, but instead it receives other special traits when the druid advances, as outlined in the Player's Handbook, page 36.
Incorrect, the animal companion gains everything according to monster progression according to several postings in the wizard site. Additionally, look up any published module from wizard that provides stats for a High EL Druid and you will note that all the animals are progressed with the appropriate modifications from the monster manual in addition to those from the druid animal companion tables.
I will try and followup with links to these post, but the first line should make the writer intent clear. "A druid's animal companion is superior to a normal animal of its kind and has special powers, as described below."
