Added: I ran the regression with total employment, and found higher significance than I did with teens. It's added below. But, it's hard to separate the cause and effect in the aggregate data.
Following up on
this, and
this, I figured at this point I could do an easy scatterplot or two to see if I could see how much the business cycle affected the changes in the employment trends here. I figured one would expect to see a range of outcomes centered around a negative mean, with the deviation in the outcomes correlating to the effect of the business cycle  with worse outcomes during recessions. Usually the correlation isn't so clean in practice. But, here it looks surprisingly uniform. And the contraction in teen EPR also has a surprisingly regular relationship with the scale of the MW hike:
Since I have simplified this into just 7 separate observations, I can see how the scale of the trend change relates to both the scale of the minimum wage hike and to the change in real GDP.
The change in the Minimum Wage, as measured, is the change, from the level the month before the first hike to the level after the last hike in the series, in the ratio of the minimum wage to the average wage.
The change in real GDP is measured over the 24 months from 1 year before the initial MW hike to 1 year after. Declining unemployment tends to lag declining GDP, and this amount of lag seems to have the strongest correlation to the scale of teen EPR in these 7 episodes.
These two independent variables don't have a relationship with each other.
The dependent variable, Teen Employment to Population Ratio, is measured here as the difference between the preMW trend slope and the trend slope during the MW episodes, accumulated over 30 months.
Below is the regression of Teen EPR against both of these variables. I was surprised to see the intercept so close to zero and such high correlations for both variables, together and separately.
Summary for Teen EPR:
MW/AW up by 10% > Teen EPR down by 10%
1 year RGDP up by 10% > Teen EPR up by 6%
Typical Episode: MW/AW up 9%, RGDP up 4% > Teen EPR down by 6.3%
ADDED: I decided to go ahead and make the same changes to the 7 observations for the total EmploymentPopulation Ratio. The coefficients are smaller, as one would expect, but the correlations are higher. I suspect that this is partly because total EPR is just a cleaner, more stable data series.
Here are the individual correlations.
As with the teen data, the independent variables are not correlated with one another. (edit: Hm...after playing with this correlation, I see that there does tend to be a correlation between lagged RGDP and the size of the MW hike if I delete the 1956 observation. So, on the one hand, there could be some colinearity here. On the other hand, declines in RGDP are correlated with the size of the MW hike. So, either the size of the MW hike is causing an RGDP decline, or, some set of factors is leading Congress to not only implement MW hikes at the end of a recovery, but to make MW hikes larger, coincidentally, when there are larger approaching economic busts. For the teen data, when I used nonlagged RGDP growth, it lowered the strength of RGDP and the regression, so using the lagged RGDP seemed reasonable. But, for total employment, using nonlagged RGDP strengthens the regression and the RGDP correlation with employment. This fits with what I had found earlier,
that the unemployment effect in the younger age groups tended to happen earlier and recover earlier than in the older age groups. In general, it may be harder to capture a clean correlation in the total employment data compared to the teen data, because MW effects on the total labor force would show up in RGDP, while it would be possible for employment changes that were focused on teens to have relatively little effect on the aggregate numbers.)
Below is the comparison of actual EPR changes during the 30 month periods of the 7 minimum wage episodes to the levels predicted by the regression coefficients.
And, below that are the regression statistics. The Fstat is .047, and the pvalues for the two coefficients are .107 for the minimum wage hike and .069 for RGDP. This is excel output, which is based on 2tailed probabilities.
Here is the data:

Emp trend change*30 months

MW/AW%ch

2 yr RGDP

1956 
4.7% 
13.0% 
6.2% 
1961 
1.0% 
7.5% 
8.8% 
1967 
1.1% 
9.4% 
6.6% 
1974 
3.9% 
10.0% 
2.0% 
1990 
3.7% 
7.5% 
1.9% 
1996 
0.7% 
5.8% 
8.7% 
2007 
5.4% 
9.4% 
2.0% 
Summary for Aggregate EPR:
MW/AW up by 10% > EPR down by 5.6%
1 year RGDP up by 10% > EPR up by 3.9%
Typical Episode: MW/AW up 9%, RGDP up 4% > EPR down by 2.7%
PS. Here is the correlation between the two independent variables.
The RGDP with a 1yr lag is what I used in the regressions. But I noticed that when I removed 1956 (the blue dot at the top right), the correlation looked similar to the correlation between the change in MW and the concurrent change in RGDP (in red). There is a slight correlation  there is actually just under a 10% pvalue on a onetailed test.
But, if the issue with minimum wage job losses was just a matter of bad timing, then, on this correlation, the trendline would be flat, but would be lower than the average 2 year RGDP growth (which is about 6.3%). What we see instead are regression lines that start out at about the average 2 year RGDP growth rate and then decline steeply as the scale of the minimum wage hike increases. This suggests that the declining RGDP is a product of the MW hikes.
The regression for EPR becomes very strong when I use the concurrent RGDP. The MW variable doesn't improve the significance, but the concurrent RGDP is very likely to be signifying some employment loss from the minimum wage. While the variance in the regression between the two independent variables is large, the effect is also large, so that the expected difference in 2 year concurrent RGDP between the smallest MW hike and the largest is more than 8%.