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I don't know if this adds anything, but the context is computer software where a component checks the health of the system and produces a report.
I don't know whether it's a 'Health Check Feature' or a 'Healthcheck Feature'.
I don't know why, but 'healthcheck' seems more natural in the sentence 'I'm running a healthcheck', whereas the the other version seems better in the sentence 'the results of the health check were...'
You should use "health check". The compound word "healthcheck" is not common.
This compound word may sound more natural to you because there is a common compound word used in this setting with almost the same meaning: "checkup".
To see why "health check" is better, try replacing those two words with two words that have similar grammatical and semantic roles: "disk test". Would you say "I'm running a disktest"? I think not.
