From Wikipedia,
Why is a private called a private in the Army?
The term derives from the medieval term "private soldiers" (a term still used in the British Army), denoting individuals who were either hired, conscripted, or mustered into service by a feudal nobleman commanding a battle group of an army. The usage of "private" dates from the 18th century.
What is a British soldier called?
Soldiers in the British Army have been called Tommies since the Boer War. But the name comes from the Duke of Wellington, who, towards the beginning of the 19th Century, used the name “Tommy Atkins” as an example of an average British soldier.
What are squaddies like?
It is an old term for an Indian irregular soldier working with the British Army in India. Squaddie - a soldier. The commonly used mis-spelling of the word 'swaddy'. This derived from the days of the army in India when swaddies were irregular Indian troops used as 'cannon-fodder' and for fatigues.
And then you have this from Stephen Tempest, a qualified amateur historian.
According to the Oxford English Dictionary, it probably derives from swad or swaddie, which is an old Northern England dialect word for "a loutish fellow", "a country bumpkin" or a "big, stout chap". That word in turn derives from Scandinavian - compare the Norwegian word svadde with the same meaning.
Citations for the basic meaning go back to the 16th century - "Dost thou drinke all thy thrift thou swilbold swadd?" is from a play written in 1577, for instance. The earliest reference to it meaning specifically a soldier is from John Hall's Memoirs written in 1708; there are other citations given from 1757 and 1787.
The folk-etymology that it comes from a Hindi word seems to be false. Lots of British Army 19th century slang came from India, but it seems this particular word was already in use long before any British swaddies set foot in India.
As Scott's answer says, the word swaddie was changed to squaddie around the time of the Second World War, since the dialect word swaddie was no longer in use outside the military and thus unfamiliar, and non-soldiers misheard it and assumed it derived from 'squad'.
Finally, note that while the British Army doesn't have a formal sub-division called a squad, the word has been in use for at least 300 years, including in official regulations, to mean "a small body of men formed together to perform a specific task". ('Firing squad', for instance.)
It's funny though, I never asked the question as to why I was called a Squaddie while I was in the army.