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In Pokémon Diamond, Pearl, Platinum, HeartGold and SoulSilver, can legendary Pokémon be encountered and caught as shiny?

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Yes.

Legendary Pokemon can be Shiny with the same rarity as any other pokemon.

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Yes.
This goes for all pokemon games:

Whenever you encounter a (wild) pokemon, the game generates it's individual values. These are numbers ranging from 0-31 for every stat and they determine how much that stat will grow when the pokemon levels up. The exact conditions for a shiny encounter can be found here but in short, based on the generated IVs the game decides if the encounter is a shiny. Since the IV generating process is the same for all pokemon (with equal odds for every species) the chance of encountering a shiny legendary is the same as a normal pokemon (although a shiny from another species might be easier to farm).

After reading through the bulbapedia, it seems I was not entirely correct, some pokemon are not possible to get shiny without glitches:
- Generation II -
Mew*
Unown forms other than I and V*
- Generation III -
Celebi
- Generation IV -
Celebi
Arceus
Spiky-eared Pichu
- Generation V -
Celebi
Arceus
Victini
Reshiram
Zekrom
Keldeo
Meloetta
Genesect

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  • Note that since Gen III, shininess has nothing to do with IVs directly, but depends on the Pokémon's internal ID (PID). The way this ID is generated in Gen IV links it to certain IV spreads, but this is no longer true in Gen V onwards. And even in Gen IV, it's only true for non-bred Pokémon.
    – scenia
    Commented Mar 4, 2014 at 11:29

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