Function `glmer()` uses by default the Laplace approximation, which is not optimal for dichotomous data. A better alternative is the adaptive Gaussian quadrature. You can use this method by setting argument `nAGQ` of `glmer()` to a higher number (e.g., 11 or 15) or alternatively using the [**GLMMadaptive**][1] package. In your example, it gives:

<!-- language-all: lang-r -->


    library("GLMMadaptive")
    helpmeobiwan <- list(NestPlot = c(1, 0, 0, 0, 0 ,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0),NumDeadJun = c( 0.1409216, -0.1932639,-0.5274494,-0.5274494, 0.1409216, -0.5274494, -0.5274494 , 0.4751071, -0.5274494 , 2.1460347 ,-0.5274494, -0.1932639, 0.8092926, -0.5274494, -0.5274494 ,-0.5274494 ,-0.1932639, 0.1409216, -0.5274494, -0.5274494 ,-0.5274494, -0.5274494 ,-0.5274494,  0.1409216,-0.5274494, -0.5274494 ,-0.5274494,  0.1409216, -0.5274494,  0.1409216, -0.5274494, -0.5274494, -0.5274494, -0.1932639, -0.1932639, -0.5274494,  0.4751071 , 0.1409216 ,-0.5274494, -0.5274494, -0.5274494, -0.5274494, -0.5274494, -0.5274494, -0.5274494, -0.5274494, -0.1932639, -0.5274494, -0.5274494 ,-0.5274494 ,-0.5274494,  0.1409216, -0.5274494, -0.5274494, -0.1932639, -0.5274494, -0.5274494, -0.5274494,  0.1409216, -0.5274494, -0.5274494  ,3.1485912 , 2.4802202,  1.4776637, -0.5274494 , 2.8144057, -0.5274494, -0.5274494,  1.1434781,  3.8169623,  3.8169623 ,-0.1932639, -0.5274494  ,1.4776637 , 1.8118492, -0.5274494),RandomPair = c(  "Madera2" ,  "Starfire1", "Madera2" ,  "Madera3" ,  "Starfire1" ,"Starfire1", "Starfire2", "Madera1" , "Madera3"   ,"Starfire2" ,"Starfire2", "Madera1",   "Madera2",  "Starfire1", "Starfire1" ,"Starfire1", "Madera1",   "Madera2" ,  "Starfire1", "Starfire1", "Starfire1", "Madera1" ,  "Starfire1", "Starfire1", "Madera1",   "Madera1" , "Starfire1", "Madera2" ,  "Madera1",   "Madera2" ,  "Madera1" ,  "Madera1"  , "Starfire1" ,"Starfire1", "Starfire1" ,"Starfire1" ,"Madera2"  , "Madera2",   "Starfire2" ,"Starfire2", "Starfire2" ,"Madera3" ,  "Madera3" ,  "Madera3" ,  "Madera3" ,  "Madera3" ,  "Starfire2", "Starfire2", "Starfire2", "Starfire2" ,"Starfire2", "Madera3",  "Madera3" ,  "Starfire2", "Madera3" ,  "Madera1"  , "Starfire2" ,"Starfire1", "Madera2" ,  "Madera3" ,  "Madera3"  , "Madera2"  , "Madera3"   ,"Starfire2", "Madera3",   "Starfire1", "Madera3"  , "Starfire2", "Starfire1", "Madera3",   "Starfire1", "Starfire2" ,"Madera1" ,  "Starfire2", "Starfire2", "Madera1"  ))
    helpmeobiwan <- as.data.frame(helpmeobiwan)

    fm <- mixed_model(NestPlot ~ NumDeadJun, random = ~ 1 | RandomPair, 
                      family = binomial(), data = helpmeobiwan)

    summary(fm)
    #> 
    #> Call:
    #> mixed_model(fixed = NestPlot ~ NumDeadJun, random = ~1 | RandomPair, 
    #>     data = helpmeobiwan, family = binomial())
    #> 
    #> Data Descriptives:
    #> Number of Observations: 76
    #> Number of Groups: 5 
    #> 
    #> Model:
    #>  family: binomial
    #>  link: logit 
    #> 
    #> Fit statistics:
    #>   log.Lik      AIC      BIC
    #>  -46.2248 98.44959 97.27791
    #> 
    #> Random effects covariance matrix:
    #>                StdDev
    #> (Intercept) 0.0477673
    #> 
    #> Fixed effects:
    #>             Estimate Std.Err z-value  p-value
    #> (Intercept)  -0.1568  0.2829 -0.5544 0.579304
    #> NumDeadJun   -1.2274  0.4917 -2.4961 0.012558
    #> 
    #> Integration:
    #> method: adaptive Gauss-Hermite quadrature rule
    #> quadrature points: 11
    #> 
    #> Optimization:
    #> method: hybrid EM and quasi-Newton
    #> converged: TRUE


  [1]: https://drizopoulos.github.io/GLMMadaptive/