Did indentured servants in British North America have job security?

Indentured servants were subject to a contract, which was generally on-sellable. ( http://www.virginiamemory.com/blogs/out_of_the_box/2013/09/11/buying-and-selling-servants/comment-page-1/ )

With sunk costs of more than £20 in 18th century money masters would be reluctant to deed or peppercorn an indenture; particularly where the servant could be sold on to a right brutal bastard who’d sweat the labour.

Historian Lerone Bennett, Jr. notes that "Masters given to flogging often did not care whether their victims were black or white." (Wikipedia “Indentured servitude” citing Calendar of State Papers: Colonial series. Great Britain. Public Record Office. 1893. p. 36)

And if you’re unwilling to accept a peppercorn you can always murder them ( https://jenjdanna.com/blog/2012/1/10/forensic-case-files-murder-of-a-colonial-servant.html )

The exception is of course apprenticeship or pseudo-apprenticeship indenture, where the master is effectively inviting someone into their family. Families, of course, can be brutally abusive and exploitative, but this is the area where early freedom and remployment as free labour is likely.

The majority of indentured servants took advantage of the labour shortage to seek better conditions as free labour than as plantation labour.

“What servants could do if they found themselves indentured in a job they really liked or that was very easy for them?” Marry him. Only in small and intimate farms, unlikely to purchase indentures due to cost and unable to compete with plantations, would an indenture show similar characteristics to an apprenticeship indenture. Redemptioners are an example of the failure of this: the family can’t afford to purchase their passage and freedom.

Finally, the following anachronisms in your question are unuseful:

• employer: you mean master. Subordination was much harder and often corporal.

• business: corporations as we know them were a 19th century phenomena.

• downsizing / layoffs: Absentee landlords or plantation owners could go bankrupt, but not through the business cycle we’re aware of. Agricultural economic cycles were slower: leases were often 20 years or more. Moreover plantations were present in an expanding market. Bankruptcy was generally a result of dissolute conduct and the labour contracts would be sold on with the real property in liquidation. More simply, labour was so short, capital (land) under-utilised, and profits so large that indentured labour would be worked to the hilt. We sack workers in our societies because of a lack of effective demand: using capital becomes unprofitable. The effective demand for tobacco was not exhausted in the 18th century. “I enjoy what I do and I'm immune to layoffs!” Given that labour discipline was customary for free labour, and corporal for indentured labour, “layoffs” are incomprehensible. The low organic composition of capital combined with an inexhaustible demand for plantation products meant that proto-capital would have an inexhaustible demand for labour: beggaring yourself in order to threaten someone with freedom doesn’t make sense. The use of physical violence to control labour meant that discipline by unemployment was unnecessary. And if you love your labourer enough to free them for a peppercorn, they’re up the duff or your family already.

• with more employer-provided training: training doesn’t exist until the 19th century. You’re thinking of instruction in trade for apprentices. And they’re not the standard unskilled agricultural labourer.

• and more tolerance of some level misbehavior on the job: it is hard to differentiate worked and unworked hours for personally owned servants. They were always on the job. The length of the working day is restricted largely by light, not labour unrest. Correspondingly the tolerance for misbehaviour is to do with different forms of labour discipline. You and I are threatened with the sack. He and she were threatened with beating and starvation.