The "margin of error" of 38% is computed using a formula for 0/1 results that are obtained independently and randomly from a large population. None of these apply. The analogous formula for the present case (with 1..7 results obtained from a small population assuming random non-response) would involve a sample standard deviation with a finite population correction. It would not be very helpful in sorting out the confusion. Instead, let's explore the concept starting from its foundations.
The purpose of a margin of error is to indicate how much the population might differ from the sample, assuming that the data are a random sample of the population.
The problem we have is we don't know what the four missing respondents would have said. We have to cover all possible cases that are consistent with the data.
An interesting possibility is that seven of the 11 CEOs would give a reply of 7, two of them would reply with 5, and the other 2 with 1: this is (hypothetically) the population. How consistent are the data with this scenario? In this case, the chance of observing six 7's and one 5 at random is
$$\frac{\binom{7}{6}\binom{2}{1}\binom{2}{0}}{\binom{11}{7}} = \frac{7 \times 2 \times 1}{330} \approx 4.24\%.$$
I imagine that observing seven 7's ($1/330$ chance) or five 7's and two 5's ($21/330$ chance) would also be considered a "very satisfied" overall rating. The total chance of observing this rating in the sample would therefore equal $(14 + 1 + 22)/330$ = $11.2\%$.
This is about the worst situation that is consistent with the recorded observations in the sense that the conclusion made from our sample ("very satisfied") has at least a 5% chance of arising from a random selection. A reasonable way to express the "margin of error," then, is to note that as many as two of the CEOs, but not more than that, might have been extremely dissatisfied.
It's a good idea to go further with the analysis, because we have no evidence to support the assumption that the seven respondents actually are a representative sample. In fact, most likely they are not. Perhaps, for example, four of the CEOs would reply with 1's but did not care to because they did not want to reveal their dissatisfaction: the missingness is not at random.
A frank and thoughtful exposition of the results would bring up both these points when assessing the credibility of the results and their applicability to the entire population. Its conclusions would necessarily be tentative. They might be stated thus:
In this self-selected sample of 7 of the 11 CEOs, six reported being "very satisfied" and one "somewhat satisfied" with their bonus. Nothing is known about what the remaining four CEOs feel. We can conclude that as a group, the majority of the 11 CEOs were highly satisfied, but there may be anywhere from none through a significant minority (four) who were dissatisfied.
One advantage of this plain exposition is that it makes no unnecessary technical demands on the reader (or the writer) by invoking a "margin of error" formula which would need to be explained and interpreted and might well be incorrect by not accounting for the small population size or possibility of non-random response.
The techniques illustrated here apply to similar problems with small population sizes, non-random responses, or qualitative ordinal scales.
Best Answer
Because it is not possible to have a percentage less than zero, the first interpretation is the response is that between 0% and 52% were happy. Just substitute 0 for the negative percentage.
The bigger question is why they are reporting results with such large margins of error. As a client, knowing that between 0% and 52% of respondents were happy is a pretty meaningless result. With such a small sample size, they are going to get those large margins of error for every single question in the survey. Ten people is just too small a sample size to get robust estimates.
For me, the bigger question is why they didn't use a qualitative method for this work.
Update based on comment below: increasing the margin of error to decrease the sample size is not a good way forward as you have seen with the margins of error you currently have. With a population of 60, you need to sample the entire population, i.e. undertaken a census, in order to get acceptably low margins of error. As for any survey, non-response bias will be a concern.
Recommendation: for the population of 60 either