A Flax plant cannot have Bonemeal applied to it if it is past its initial stage of growth. This crop is similar to Sugar Cane in regards to harvesting: as long as the bottom portion of the plant remains intact, the plant will regrow.
Harvesting Flax without a Sickle will cause the plant to drop approximately 2-3 String and 2-3 Flax Seeds. Harvesting with a Sickle will cause the Flax to drop between 3-4 Flax Seeds and around 4-5 String. Using Bonemeal and Flax Seeds, it will cost 52 EMC, while harvesting with a Sickle will give a return of 2-3 Flax Seeds (as one of the seeds is required to replant) and 4-5 String. At the minimum, this will give 8-12 EMC through the seeds, and 48-60 EMC through the String. Therefore, farming Flax with Bonemeal and a Sickle will guarantee a profit.