Some observations

1. This could be a numerical issue. Floating point arithmetic has finite precision, so values $\epsilon$ and $1 - \epsilon$ for some small $\epsilon >0$ may be rounded to 0 and 1. 

2. This could be a printing precision issue. When displaying these values, you may only be printing the first several digits, and when the values are very close to 0 and 1, the result is rounded to 1 and 0.

3. It could be that the weights are so large that the inputs to the softmax activation are saturated very rapidly. You can bound weights away from large values using $L^2$ regularization.

4. This is tangential to the question you asked, but dropout and batch norm don't work well together. 

  Günter Klambauer, Thomas Unterthiner, Andreas Mayr and Sepp Hochreiter. "Self-Normalizing Neural Networks"

  > To robustly train very deep CNNs, batch normalization evolved into a standard to normalize neuron activations to zero mean and unit variance [17]. Layer normalization [1] also ensures zero mean and unit variance, while weight normalization [25] ensures zero mean and unit variance if in the previous layer the activations have zero mean and unit variance. Natural neural networks [6] also aim at normalizing the variance of activations by reparametrization of the weights. However, training with normalization techniques is perturbed by stochastic gradient descent (SGD), stochastic regularization (like dropout), and the estimation of the normalization parameters. Both RNNs and CNNs can stabilize learning via weight sharing, therefore they are less prone to these perturbations. In contrast, FNNs trained with normalization techniques suffer from these perturbations and have high variance in the training error (see Figure 1). This high variance hinders learning and slows it down. Furthermore, strong regularization, such as dropout, is not possible as it would further increase the variance which in turn would lead to divergence of the learning process. We believe that this sensitivity to perturbations is the reason that FNNs are less successful than RNNs and CNNs.

  My recommendation is to use either dropout or batch normalization; alternatively, you may wish to implement a network using SELUs and alpha dropout as suggested in this paper.

5. Training a neural network is hard. Here are some general suggestions. https://stats.stackexchange.com/questions/352036/what-should-i-do-when-my-neural-network-doesnt-learn/352037#352037