The short answer is no, there were not white slaves in either the United States or the Caribbean. This notion has developed a little bit out of genuine historical debate, but it is driven mainly by ignorance and by people advancing an agenda. The misconception arises because of the existence of indentured servitude - another form of coercive labour whereby white residents of Britain signed up (not always particularly voluntarily) to work in British colonies as labourers for a fixed period of time, with the promise of land or cash at the end of their term. This was initially the preferred form of coercive labour in the colonies, and it certainly could be brutal. The planter class had considerable control over their servants and brutality was the norm, not the exception; many of the systems and institutions that served to underpin slavery have their initial roots in the practice of indentured labour. I discussed this in this answer that you might find interesting.

As such, there has been some debate in the past as to whether or not indentured servitude could be considered a kind of white slavery. The general consensus is that it is not. However appalling conditions could be for servants, they remained British subjects and members of society; they were not degraded in the complete and dehumanising fashion that African slaves were. Their legal rights were significantly stronger, even if the extent to which they were upheld was patchy, and they did not experience the kind of complete 'social death' of slavery that Orlando Patterson described. Crucially, they remained theoretically free persons who were subject themselves to temporary servitude on the promise of economic gain - slaves were, by definition, held until death or manumission, and if they did gain their freedom, they did not have any kind of material reward waiting for them. Indentured servants on the other hand had the potential not only to rejoin civil society after their term of service, but to establish themselves in it as a person of respect and prominence.

Less clear is the relationship between this practice and slavery, which also sometimes serves to fuel this myth of white slavery. Some have suggested for instance that African slavery grew out of indentured servitude, with the first black slaves being indentured servants on par with white servants. Others, myself included, hold the position that whilst some black individuals may have initially been indentured servants, from the outset Africans were generally held as slaves even if the law had not yet developed to describe and regulate this phenomenon (this related answer may interest you on this point). For some individuals advancing an agenda or misinterpreting history, the notion that African slavery may have grown out of indentured servitude seems to reinforce the idea that white servants were also slaves, but that is not the implication of the idea.

A genuine argument can be made for comparing white indenture to slavery, but the consensus is generally that this is inappropriate and fails to capture the nuanced reality of the situation. There are simply too many significant, fundamental differences between the two practices to consider them one in the same, and even those who make genuine contentions that white indenture is a form of slavery are generally arguing from a broader, comparative perspective on systems of unfree and coercive labour - they are not trying to suggest that white indenture diminishes the experience, significance or severity of black slavery, as populist 'white slavery' advocates often set out to do.