The verb to lay is a transitive verb, which means it requires an object (such as an egg). In this case the word 'low' could be substituted for any prepositional phrase, such as in the straw.
Future: 'The hen is going to lay an egg low.'
Present: 'The hen is laying an egg low.'
Past: 'The hen laid an egg low.'
Past perfect: 'The hen had laid an egg low.'
The confusion between lie low and lay low stems from the fact that the past tense of to lie (intransitive verb) is lay; whereas the past tense of to lay (transitive verb) is laid. Also, in this case lie low is an idiom so both words must be used together; this is consistent with many other idioms, such as 'hurry up,' for example.
When hiding out, the verb to lie is the appropriate choice. Similarly, the verb to lie is the correct choice for 'going to lie down,' which has the identical pattern of verb tense usage as going to lie low for exactly the same reasons.